Three subgroups lemma

In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of the Hall–Witt identity.

Notation

In that which follows, the following notation will be employed:

• If H and K are subgroups of a group G, the commutator of H and K will be denoted by [H,K]; if L is a third subgroup, the convention that [H,K,L] = [[H,K],L] will be followed.
• If x and y are elements of a group G, the conjugate of x by y will be denoted by $x^{{y}}$.
• If H is a subgroup of a group G, then the centralizer of H in G will be denoted by C_G(H).

Statement

Let X, Y and Z be subgroups of a group G, and assume

$[X,Y,Z]=1$ and $[Y,Z,X]=1$

Then $[Z,X,Y]=1$.^[1]

More generally, if $N\triangleleft G$, then if $[X,Y,Z]\subseteq N$ and $[Y,Z,X]\subseteq N$, then $[Z,X,Y]\subseteq N$.^[2]

Proof and the Hall–Witt identity

Hall–Witt identity

If $x,y,z\in G$, then

${\displaystyle [x,y^{-1},z]^{y}\cdot [y,z^{-1},x]^{z}\cdot [z,x^{-1},y]^{x}=1.}$

Proof of the three subgroups lemma

Let $x\in X$, $y\in Y$, and $z\in Z$. Then $[x,y^{{-1}},z]=1=[y,z^{{-1}},x]$, and by the Hall–Witt identity above, it follows that $[z,x^{{-1}},y]^{{x}}=1$ and so $[z,x^{{-1}},y]=1$. Therefore, $[z,x^{{-1}}]\subseteq {\mathbf {C}}_{G}(Y)$ for all $z\in Z$ and $x\in X$. Since these elements generate $[Z,X]$, we conclude that $[Z,X]\subseteq {\mathbf {C}}_{G}(Y)$ and hence $[Z,X,Y]=1$.

See also

• Commutator
• Lower central series
• Grün's lemma
• Jacobi identity

Notes

1. Isaacs, Lemma 8.27, p. 111
2. Isaacs, Corollary 8.28, p. 111

References

• I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.